×

Immersed finite element method. (English) Zbl 1067.76576

Summary: The immersed finite element method (IFEM) is proposed for the solution of complex fluid and deformable structure interaction problems encountered in many physical models. In IFEM, a Lagrangian solid mesh moves on top of a background Eulerian fluid mesh which spans over the entire computational domain. Hence, the mesh generation is greatly simplified. Moreover, both fluid and solid domains are modeled with the finite element methods and the continuity between the fluid and solid sub-domains is enforced via the interpolation of the velocities and the distribution of the forces with the reproducing kernel particle method (RKPM) delta function. In comparison with the immersed boundary (IB) method, the higher-ordered RKPM delta function enables the fluid domain to have nonuniform spatial meshes with arbitrary geometries and boundary conditions. The use of such kernel functions may eventually open doors to multi-scale and multi-resolution modelings of complex fluid-structure interaction problems. Rigid and deformable spheres dropping in channels are simulated to demonstrate the unique capabilities of the proposed method. The results compare well with the experimental data. To the authors’ knowledge, these are the first solutions that deal with particulate flows with very flexible solids.

MSC:

76M10 Finite element methods applied to problems in fluid mechanics
76T20 Suspensions
74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Clift, R.; Grace, J. R.; Weber, M. E., Bubbles, Drops, and Particles (1978), Academic Press
[2] Peskin, C. S., Numerical analysis of blood flow in the heart, J. Comput. Phys., 25, 220-252 (1977) · Zbl 0403.76100
[3] Peskin, C. S., The immersed boundary method, Acta Numer., 11, 479-517 (2002) · Zbl 1123.74309
[4] Glowinski, R.; Pan, T. W.; Hesla, T. I.; Joseph, D. D.; Périaux, J., A fictitious domain approach to the direct numerical simulation of incompressible viscous flow past moving rigid bodies: application to particulate flow, J. Comput. Phys., 169, 363-426 (2001) · Zbl 1047.76097
[5] Hu, H. H.; Patankar, N. A.; Zhu, M. Y., Direct numerical simulations of fluid-solid systems using the arbitrary Lagrangian-Eulerian technique, J. Comput. Phys., 169, 427-462 (2001) · Zbl 1047.76571
[6] Huerta, A.; Liu, W. K., Viscous flow with large free surface motion, Comput. Methods Appl. Mech. Engrg., 69, 277-324 (1988) · Zbl 0655.76032
[7] Johnson, A.; Tezduyar, T. E., Advanced mesh generation and update methods for 3d flow simulations, Comput. Mech., 23, 130-143 (1999) · Zbl 0949.76049
[8] N.A. Patankar, X. Wang, W.K. Liu, A note on immersed boundary formulation for elastic bodies in fluids, preprint, 2003; N.A. Patankar, X. Wang, W.K. Liu, A note on immersed boundary formulation for elastic bodies in fluids, preprint, 2003
[9] Saad, Y.; Schultz, M. H., GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Statist. Comput., 7, 3, 856-869 (1986) · Zbl 0599.65018
[10] Belytschko, T. B.; Liu, W. K.; Moran, B., Nonlinear Finite Elements for Continua and Structures (2000), John Wiley & Sons
[11] Tezduyar, T. E., Stabilized finite element formulations for incompressible-flow computations, Adv. Appl. Mech., 28, 1-44 (1992) · Zbl 0747.76069
[12] Tezduyar, T. E., Finite element methods for flow problems with moving boundaries and interfaces, Arch. Comput. Methods Engrg., 8, 2, 83-130 (2001) · Zbl 1039.76037
[13] Hughes, T. J.R.; Franca, L. P.; Balestra, M., A new finite element formulation for computational fluid dynamics: V. Circumventing the Babuška-Brezzi condition: a stable Petrov-Galerkin formulation of the Stokes problem accommodating equal-order interpolations, Comput. Methods Appl. Mech. Engrg., 59, 85-99 (1986) · Zbl 0622.76077
[14] Hughes, T. J.R.; Liu, W. K.; Zimmerman, T. K., Lagrangian-Eulerian finite element formulations for incompressible viscous flows, Comput. Methods Appl. Mech. Engrg., 29, 329-349 (1981) · Zbl 0482.76039
[15] Wang, X., On the discretized delta function and force calculation in the immersed boundary method, (Bathe, K. J., Computational Fluid and Solid Mechanics (2003), Elsevier), 2164-2169
[16] X. Wang, W.K. Liu, Extended immersed boundary method using FEM and RKPM, Comput. Methods Appl. Mech. Engrg., in press; X. Wang, W.K. Liu, Extended immersed boundary method using FEM and RKPM, Comput. Methods Appl. Mech. Engrg., in press · Zbl 1060.74676
[17] Liu, W. K., Finite element procedures for fluid-structure interactions and application to liquid storage tanks, Nucl. Engrg. Des., 65, 221-238 (1981)
[18] Liu, W. K.; Gvildys, J., Fluid-structure interaction of tanks with an eccentric core barrel, Comput. Methods Appl. Mech. Engrg., 58, 51-77 (1986) · Zbl 0595.73045
[19] Liu, W. K.; Jun, S.; Zhang, Y. F., Reproducing kernel particle methods, Int. J. Numer. Methods Fluids, 20, 1081-1106 (1995) · Zbl 0881.76072
[20] Liu, W. K.; Ma, D. C., Computer implementation aspects for fluid-structure interaction problems, Comput. Methods Appl. Mech. Engrg., 31, 129-148 (1982) · Zbl 0478.73061
[21] Zhang, L.; Wagner, G. J.; Liu, W. K., A parallaized meshfree method with boundary enrichment for large-scale CFD, J. Comput. Phys., 176, 483-506 (2002) · Zbl 1060.76096
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.